L-functions and Automorphic Representations James Arthur Abstract. Our goal is to formulate a theorem that is part of a recent classification of automorphic representations of orthogonal and symplectic groups. To place it in perspective, we devote much of the paper to a historical introduction to the Langlands program. In our attempt to make the article accessible to a general mathematical audience, we have centred it around the theory of L-functions, and its implicit foundation, Langlands principle of functoriality. Mathematics Subject Classification (2010). Primary 11F66, 11F70, 11R37, 11F57, 22F55. Keywords. L-functions, automorphic representations, functoriality, classical groups, discrete spectrum. Preface Suppose that f(x) is a monic polynomial of degree n with integral coefficients. For any prime number p, we can then write f(x) as a product f(x) f 1 (x) f r (x) (mod p) of irreducible factors modulo p. It is customary to leave aside the finite set S of primes for which these factors are not all distinct. For those that remain, we consider the mapping p Π p = {n 1,..., n r }, n i = deg ( f i (x) ), from primes p / S to partitions Π p of n. Here are two basic questions: (I) Is there some independent way to characterize the preimage of any partition Π of n? P(Π) = {p / S : Π p = Π} (II) What is the density of P(Π) in the set of all primes, or for that matter, the set of all positive integers. Supported in part by NSERC Discovery Grant A3483.
2 James Arthur Suppose for example that f(x) = x 2 + 1. Then S consists of the single prime 2, while P(1, 1) = {p : p 1 (mod 4)} and P(2) = {p : p 3 (mod 4)}. This well known supplement of the law of quadratic reciprocity gives us a striking answer to (I). As for (II), P(1, 1) and P(2) are each known to have density 1 2 in the set of all primes. Combined with the prime number theorem, this gives an asymptotic formula ( x ) lim P(Π, x) / = 1/2, Π = {1, 1}, {2}, x log x for the set P(Π, x) = {p P(Π) : p x} of primes p P(Π) with p x. If the generalized Riemann hypothesis can be proved, there will be much sharper asymptotic estimates. In general, the two questions have significance that goes well beyond their obvious initial interest. The first could perhaps be regarded as the fundamental problem in algebraic number theory. The second has similar standing in the area of analytic number theory. Both questions are central to the Langlands program. In this article we combine an introduction to the theory of automorphic forms with a brief description of a recent development in the area. I would like to make the discussion as comprehensible as I can to a general mathematical audience. The theory of automorphic forms is often seen as impenetrable. Although the situation may be changing, the aims and techniques of the subject are still some distance from the common mathematical canon. At the suggestion of Bill Casselman, I have tried to present the subject from the perspective of the theory of L-functions. These are concrete, appealing objects, whose behaviour reflects the fundamental questions in the subject. I will use them to illustrate the basic tenets of the Langlands program. As we shall see in 4, L-functions are particularly relevant to the principle of functoriality, which can be regarded as a foundation of the Langlands program. The new development is a classification [A4] of automorphic representations of classical groups G, specifically orthogonal and symplectic groups, in terms of those of general linear groups GL(N). It was established by a multifaceted comparison of trace formulas. These are the trace formula for G [A1] and its stabilization [A2], which is now unconditional thanks to the proof of the fundamental lemma [N], and the twisted trace formula for GL(N) [LW] and its stabilization, which is still under construction [W1], [W2], [MW2]. We refer the reader to the surveys [A3], [A5], and [A6], each written from a different perspective, for a detailed description of the classification. We shall be content here to formulate a consequence of the classification in terms of our two themes, L-functions and the principle of functoriality.
L-functions and Automorphic Representations 3 Upon reflection after its completion, I observe that the article is not typical of plenary reports for an ICM. It represents a broader, and perhaps denser, introduction than is customary. I hope that the nonspecialist for whom the article is intended will find the details comprehensive enough without being overwhelming. 1. Classical introduction Recall that a Dirichlet series is an infinite series n=1 a n n s, (1.1) for a complex variable s and complex coefficients a n. If the coefficients have moderate growth, the series converges when s lies in some right half plane in C. If they are bounded, for example, the series converges absolutely when the real part Re(s) of s is greater than 1. More generally, if the coefficients satisfy a bound a n cn k, for positive real numbers c and k, the series converges absolutely in the right half plane Re(s) > k + 1. Since the convergence is uniform in any smaller right half plane, the sum of the series is an analytic function of s on the open set Re(s) > k+1. The most famous example is the Riemann zeta function ζ(s) = n=1 1 n s. (1.2) Since the coefficients are all equal to 1, this converges as an analytic function of s for Re(s) > 1. Euler had studied the series earlier for real values of s. He discovered a remarkable formula ζ(s) = ( 1 ) 1 p s (1.3) p for ζ(s) as a product over all prime numbers p, which he proved using only the fundamental theorem of arithmetic and the formula for the sum of a geometric series. With the later theory of complex analysis, Riemann was able to extend the domain. He showed that the function has analytic continuation to a meromorphic function of s in the entire complex plane, which satisfies a functional equation ζ(s) = 2 s π s 1 sin(πs/2)γ(1 s)ζ(1 s) (1.4) in terms of its values at points s and 1 s. He observed further that if L R (s) = π s/2 Γ(s/2),
4 James Arthur where Γ( ) is the gamma function (here and in (1.4)), the product L(s) = L R (s)ζ(s) = L R (s) ( 1 ) 1 p s p (1.5) satisfies the symmetric functional equation L(s) = L(1 s). (1.6) The functions ζ(s) and L(s) are both analytic in the complex plane, except for a simple pole at s = 1. Riemann conjectured that the only zeros of L(s) lie on the vertical line Re(s) = 1/2. This is the famous Riemann hypothesis, regarded by many as the most important unsolved problem in mathematics. Its interest stems from the fact that the zeros {ρ = 1/2 + it} of L(s) on this line are in some sense dual to prime numbers, or more accurately, to logarithms {γ = log p n } of prime powers. We can think of the former as a set of spectral data and the latter as a set of geometric data, which are related to each other by a Fourier transform. The Riemann hypothesis implies a very sharp asymptotic estimate for the number π(x) = P(x) = {p x} of primes less than or equal to x. One particularly explicit form [Schoen] of the estimate is π(x) li(x) 1 x log x, 8π x 2658, (1.7) for the principle value integral li(x) = x 0 1 log t dt. Because the function li(x) is easy to approximate for large values of x, and is x asymptotic to a function that strongly dominates the error term, this is a log x striking estimate indeed. For example, if x = 10 100, one sees that π(x) and li(x) are positive integers with 97 digits each, the first 47 of which match! Dirichlet later generalized Riemann s construction to series of the form L N (s, χ) = n=1 χ(n) n s, (1.8) where χ(n) is what later became known as a Dirichlet character. We recall that χ is a complex valued function on N such that χ(nm) = χ(n)χ(m), and χ(n + N) = χ(n), χ(n) = 0, if gcd(n, N) > 1,
L-functions and Automorphic Representations 5 where N is a positive integer called the modulus of χ. One says that χ is primitive if its nonzero values are not given by restriction of a Dirichlet character with modulus a proper divisor of N (in which case N is called the proper divisor of χ). This means that x generates the cyclic group of characters on the multiplicative group (Z/ZN). The series (1.8) behaves very much like the Riemann zeta function. It converges if Re(s) > 1. It has an Euler product L N (s, χ) = p ( 1 ) 1 χ(p)p s. (1.9) It also has analytic continuation to a meromorphic function on the complex plane. However, its functional equation is a little more interesting. To state it, we can suppose without loss of generality that χ is primitive. We account for the gamma function and powers of π in the analogue of (1.4) by setting L R (s, χ) = (π) (s+a)/2 Γ ( (s + a)/2 ), where a = a(χ) = { 0, if χ( 1) = 1 1, if χ( 1) = 1. The product L(s, χ) = L R (s, χ)l N (s, χ) = Γ R (s, χ) then satisfies the functional equation where χ is the complex conjugate of χ, and for ε(χ) = ε(1/2, χ) = i a N 1/2( N 1 p 1 1 χ(p)p s (1.10) L(s, χ) = ε(s, χ)l(1 s, χ), (1.11) ε(s, χ) = N 1/2 ε ε(χ), (1.12) n=1 χ(n)e 2πin/N ), i = 1. The expression in the brackets is a Gauss sum. It is the analogue for a finite field (or ring) of the classical gamma function for the field R. Its generalizations are an important part of the functional equations of nonabelian L-functions. Dirichlet introduced his L-series to study the prime numbers in an arithmetic progression. Suppose that χ is primitive and nontrivial. Dirichlet showed that the L-function (1.8) (or its normalization (1.10)) is actually an entire function of s C, and in addition, that its value L(1, χ) at 1 is nonzero. He then used this to show that for any integer a with gcd(a, N) = 1, the number π(a, N) = P(a, N) = {p a(mod N)}
6 James Arthur of primes p in the arithmetic sequence a, a + N, a + 2N,.... is infinite. The generalized Riemann hypothesis is the assertion that the only zeros of the entire function L(s, χ) again lie on the line Re(s) = 1 2. It implies an analogue π(x, a, N) φ(n) 1 li(x) < C x log x, x 2, of the asymptotic estimate (1.7), for the number π(x, a, N) = {p P(a, N) : p x} of primes in the arithmetic sequence less than or equal to x. The familiar Euler function φ(n) = {a : 1 a N, gcd(a, N) = 1} equals the number of such arithmetic progressions. These remarks illustrate the power of L-functions. They are directed at some of the deepest analytic questions on the distribution of prime numbers. The L- functions we have described are just the beginning. They are the simplest among an enormous but unified collection of L-functions, which have the potential to resolve fundamental arithmetic questions about prime numbers, as well as refinements of the analytic questions we have looked at. 2. Artin L-functions and class field theory There seems not to be complete agreement on what kind of Dirichlet series (1.1) should be called an L-function. To qualify, it should certainly converge to an analytic function in some right half plane Re(s) > k + 1. It should also have an Euler product (1 + c p,1 p s + c p,2 p 2s + ), Re(s) > k + 1, (2.1) p for a family of coefficients C = {c p,n }. Finally, it ought to have (or at least be expected to have) analytic continuation and functional equation. We shall take these conditions as our working definition. The Euler product often arises naturally as an incomplete product, taken over the primes outside a finite set S. However, the expected functional equation is generally best stated for the completed product, in which one adds factors L p (s, C) for the primes p S, as well as a factor L R (s, C) for the archimedean prime R as above. If we also write L p (s, C) for the factors with p S, we then have the given incomplete L-function L S (s, C) = p S L p (s, C),
L-functions and Automorphic Representations 7 and its better behaved completion ( L(s, C) = L R (s, C) p S ) L p (s, C) L S (s, C). (2.2) Treating Dirichlet L-functions as a guide, we would be looking for a functional equation L(s, C) = ε(s, C)L(1 s, C ), (2.3) where C is some dual family of coefficients attached naturally to C, and ε(s, C) = b (1/2 s) C ε( 1 2, C), for a positive real number b C, and a complex number ε( 1 2, C) that is independent of s. The higher L-functions that will be our topic are of two kinds, automorphic and arithmetic. The former are primarily analytic objects, while the latter are algebraic. The Riemann zeta function is the common ancestor of them all. It is a mainstay of analytic number theory. However, it can also be regarded as the trivial case of the arithmetic L-functions we shall describe in this section. Dirichlet L-functions L(s, χ) are automorphic. As in the case of the zeta function, it is the application of analysis (real, complex and harmonic) to L(s, χ) that leads to its analytic continuation and functional equation, and to the location of any poles. An important family of arithmetic L-functions was introduced by Emile Artin. These are attached to N-dimensional representations r : Γ E/Q GL(N, C) of the Galois group Γ E/Q = Gal(E/Q) of a finite Galois extension E of Q. We shall describe them here from the perspective of the questions (I) and (II) raised in the preface. For any E, we have the finite set S = S E of prime numbers p that ramify in E, and for any unramified p S E, a canonical conjugacy class Frob p (the Frobenius class) in Γ E/Q. This is one of the first constructions encountered in algebraic number theory. To be concrete, we can take E to be the splitting field of a monic, integral polynomial f(x) of degree n, as in the preface. For any prime number p, we then have the corresponding factorization of f(x) into irreducible factors f i (x) modulo p, with degrees n i. This embeds S E in the finite set S = S f of primes p for which these factors are distinct. The choice of f(x) also identifies Γ E/Q with a conjugacy class of subgroups of the symmetric group S n. For any unramified prime p / S E, Frob p is then mapped to the conjugacy class in S n defined by the partition Π p = {n 1,..., n r } of n. In particular, the set Spl(E/Q) = {p / S : Frob p = 1} of prime numbers that split completely in E, is the set of p such that f(x) breaks into linear factors modulo p. It is known [T2, p. 165] that Spl(E/Q) characterizes
8 James Arthur E. In other words, the mapping E Spl(E/Q), from Galois extensions of Q to subsets of prime numbers, is injective. A variant of the question (I) would be to characterize its image. This would amount to a classification of Galois extensions E of Q. For any p / S, the image r(frob p ) under the representation r of Γ E/Q gives a semisimple conjugacy class in the complex general linear group GL(N, C). Artin defined its local L-factor L p (s, r) = det ( 1 r(frob p )p s) 1, p S, (2.4) in terms of the associated characteristic polynomial. It clearly has an expansion in terms of powers of p s, and therefore has the general form of the factor of p in (2.1). Artin then defined an incomplete L-function as the Euler product L S (s, r) = p S L p (s, r), (2.5) which he conjectured had analytic continuation with functional equation of the general form (2.3). The question (II) will be reflected in the analytic properties of the resulting function of s. However, the analytic continuation and functional equation of (2.5), let alone the relevant analogue of the Riemann hypothesis, is a more serious proposition. Since L S (s, r) is a fundamentally arithmetic object, it cannot be studied in terms of the kind of analysis that Dirichlet applied to the L-functions L(s, χ). Artin treated it indirectly. Suppose that the Galois group Gal(E/Q) is abelian, and that r is irreducible, and therefore one-dimensional. The classes r(frob p ) are then just nonzero complex numbers. The (incomplete) Artin L-function becomes a product L S (s, r) = p S 1 1 r(frob p )p s that resembles the Euler product (1.9) of a Dirchlet L-function. Indeed, if p divides the modulus of χ (written p N), χ(p) vanishes, and the corresponding product (1.9) can then be written L S (s, χ) = 1 1 χ(p)p s, p S where S = {p : p N}. This formal similarity between the products L S (s, r) and L S (s, χ) turns out in fact to be an identity. More precisely, for any one-dimensional Galois representation r, there is a Dirichlet character χ such that the function L S (s, r) equals L S (s, χ). The new L-function therefore has the analytic behaviour of the Dirichlet L-function L S (s, χ). In particular, it has analytic continuation, and its completed L-function
L-functions and Automorphic Representations 9 (2.2) (with r in place of C) satisfies the functional equation (2.3) (with r = r in place of C ). The assertion that for every r there is a χ is a rather deep classical result, known as the Kronecker-Weber theorem. It finesses the question of the behaviour of abelian Artin L-functions by imposing limits on the set of abelian extensions of Q. From an elementary calculation in algebraic number theory, one obtains the converse theorem that for every χ there is an r. More precisely, if χ has modulus N, there is a one-dimensional representation r of the abelian Galois group of the cyclotomic Galois extension Q(e 2πi/N ) of Q such that χ(p) equals r(frob p ), for any p that does not divide N. The Kronecker-Weber theorem can then be interpreted as the assertion that any finite abelian extension of Q is contained in the cyclotomic extension of Nth roots of 1, for some N. The Kronecker-Weber theorem predated Artin by many years. However, Artin was working over an arbitrary number field F, a finite field extension of Q, rather than Q itself. The definitions we have made so far are easily extended from Q to F. The ring o = o F of algebraic integers of F does not generally have unique factorization, so one must replace prime numbers p in Z with prime ideals p in o, and integers n in Z with general (integral) ideals a in o. Any ideal then has a norm where Na = o/a = (Np 1 ) a1 (Np r ) ar = o/p 1 a1 o/p r ar, (2.6) a = p a1 1 par r is its unique factorization into prime ideals. The definition of a Dirichlet series (1.1) for Q can then be extended to F by replacing the sum over n by a sum over a, and the corresponding variable n s by (Na) s. The same goes for an Euler product (2.1). A Dirichlet series for F does reduce to a Dirichlet series for Q, since the norm of a prime ideal p is a power of a prime number p. However, a Dirichlet or Artin L-function over F represents something different, even though it can be regarded as a Dirichlet series (1.1) with Euler product (2.1). Artin worked from the beginning over F. He defined the L-function L S (s, r) for any N-dimensional representation r : Γ E/F GL(N, C) (2.7) of the Galois group of a finite Galois extension E/F. Algebraic number theory again tells us that for any p outside the finite set S = S E of prime ideals for F that ramify in E, there is a canonical conjugacy class Frob p = Frob E/F,p in Γ E/F. The definition (2.4) therefore does extend to F if we replace p s by Np s. Artin also introduced factors for the ramified primes p S, and for the archimedean primes v for F (now a finite set S rather than just the one completion R of Q). He conjectured that the resulting product L(s, r) had functional equation (2.3), with an ε-factor ε(s, r) he formulated in terms of r. It was in this context that Artin obtained the analytic continuation and functional equation for the abelian L-functions L(s, r). Dirichlet characters χ can still be defined by a variant of the prescription for Q above, and the same analysis that
10 James Arthur works for Q gives the analytic continuation and functional equation of a general Dirichlet L-function L(s, χ). With this in mind, Artin established an F -analogue of the Kronecker-Weber theorem, known now as the Artin reciprocity law. It again asserts that for any one-dimensional Galois representation r over F, there is a Dirichlet character χ over F such that χ(p) equals r(frob p ), for every unramified prime ideal p for F. This leads to the identity L(s, r) = L(s, χ) of L-functions, and therefore the desired analytic properties of the arithmetic L-function L(s, r). The Artin reciprocity law is one of the central assertions of class field theory. Unlike the general constructions above, it becomes much deeper in the passage from Q to F, even though the assertion remains similar. As in the case F = Q, the general law asserts that the abelian field extensions over F are limited to those attached to Dirichlet characters χ. These may then be classified by the reciprocity law according again to the remark on [T2, p. 165]. r(frob p ) = χ(p), p S, (2.8) For completeness, we note that Artin L-functions are but the simplest in the general family of arithmetic L-functions, called motivic L-functions. A Q-motive M over F (which we will not try to define!) also comes with a finite dimensional representation r M,l of the absolute Galois group Γ F = Γ F /F. In this general case, however, it takes values in an l-adic general linear group GL(N, Q l ), for a variable prime number l / S. It therefore gives rise to a (continuous) homomorphism r M = l r M,l : Γ F GL(N, Q l ) (2.9) l from Γ F to a large, totally disconnected group. It therefore represents a much larger quotient of Γ F than does a complex valued representation (2.7). The motive should also come with a finite set S of prime ideals in F such that r M,l is unramified in any prime p / S S l, where S l is the set of primes in F that divide the prime l of Q. The family {r M,l } of l-adic representations is conjectured to be compatible, in the sense that the image r M,l (Frob p ) of the associated Frobenius class in GL(N, Q l ) is the image of a semisimple conjugacy class r M (Frob p ) in GL(N, Q) that is independent of l. The incomplete L-function of M is then given by an Euler product L S (s, M) = p S det ( 1 r M (Frob p )(Np) s) 1, (2.10) which converges in some right half plane, and which is again expected to have analytic continuation with functional equation. Notice that any complex representation (2.7) of Γ E/F that is defined over Q gives a compatible family (2.9) of l-adic representations. It represents a Q-motive over F of dimension 0.
L-functions and Automorphic Representations 11 3. Automorphic L-functions Dirichlet L-functions are the analytic counterparts of abelian Artin L-functions. Class field theory, the culmination of many decades of effort by number theorists past, represents a classification of the finite abelian field extensions of any number field. It tells us that any abelian Artin L-function is a Dirichlet L-function. An L-function of the former sort therefore inherits all of the rich properties that can be made available for the latter through analysis. What are the analytic counterparts of nonabelian Artin L-functions? They are the automorphic L-functions introduced by Robert Langlands [L1] in 1970. Automorphic representations are the nonabelian generalizations of Dirichlet characters, and their abelian generalizations introduced later by Erich Hecke. They are defined for any connected, reductive algebraic group G over the number field F. Algebraic groups represent a conceptual hurdle for many, but a reader is invited to take G to be the general linear group GL(N) of invertible (N N)-matrices. For any ring A (abelian, with 1), G(A) is then equal to the group GL(N, A) of (N N)-matrices with entries in A and determinant equal to a unit in A. We want to take A to be the ring A = A F of adeles of F. This is a locally compact topological ring, in which F embeds as a discrete subring. It is often a second hurdle, but avoiding it would make matters considerably more complicated. The idea is really quite simple and natural. By definition, the group of adelic points of G is a restricted direct product G(A) = G(F v ), (3.1) v taken over the valuations v on F. For any v, F v is the locally compact field obtained by completing F with respect to v. It is modeled on the standard case of the completion F v = R of F = Q with respect to the usual absolute value v =. We recall that the complementary valuations for F = Q are the nonnegative functions { p r, if u = (a/b)p r, for a, b, r Z, (a, p) = (b, p) = 1, u p = 0, if u = 0, on Q, parametrized by prime numbers p. In general, the restricted direct product is the subgroup of elements x = v x v, x v G(F v ), in the direct product such that for almost all valuations v, x v lies in the maximal compact subgroup G(o v ) of points in G(F v ) with values in the compact subring o v = {u v F v : u v v 1} of integers in F v. It becomes a locally compact group under the appropriate direct limit topology. The group G(F ) embeds in G(F v ) (as a dense subgroup). The
12 James Arthur diagonal embedding of G(F ) into G(A) then exists (because an element in G(F ) is integral at almost all valuations v), and is easily seen to have discrete image. Since G(F ) is discrete in G(A), the quotient G(F )\G(A) is a reasonable object. It comes with a right invariant measure, which is determined up to a positive multiplicative constant. One can therefore form the associated space L 2( G(F )\G(A) ) of square-integrable functions. It is a Hilbert space, equipped with the unitary representation ( R(y)φ ) (x) = φ(xy), x, y G(A), φ L 2 ( G(F )\G(A) ), of G(A) by right translation. One could describe an automorphic representation of G to be an irreducible representation of G(A) that occurs in the spectral decomposition of R. This description is actually more of an informal characterization than a definition. It is also more restrictive than the formal definition in [BJ] and [L4]. We shall take the broader definition, without recalling its two equivalent formulations established in [L4]. We will then call automorphic representations that satisfy the narrower spectral condition above globally tempered. Suppose for example that G is the abelian algebraic group GL(1). Then G(A) is the multiplicative group A of elements x in A whose components x v F v are all nonzero and of valuation 1 for almost all v. This is the group of ideles, introduced earlier by Chevalley. A (globally tempered) automorphic representation of G = GL(1) is a character χ on the idele class group F \A, or in other words, a continuous F -invariant homomorphism from A to the group U(1) of complex numbers of absolute value 1. It is the generalization of a Dirichlet character introduced by Hecke, which he called a Grossencharakter, and which is now generally referred to as a Hecke character. Hecke worked in the classical context of ideals a, but his constructions are a little easier to formulate now in the language of ideles. It is an early illustration of the convenience of the language of adeles. In this regard, a Dirichlet character is just a Hecke character of finite order. One of the remarkable discoveries of Langlands has been the fundamental role played by a certain dual group of G. The dual group is a complex connected reductive group Ĝ, whose Coxeter-Dynkin diagram is the dual of the diagram of G. It comes with an action of the absolute Galois group Γ F = Γ F /F, a compact totally disconnected group, which factors through the finite quotient Γ E/F of Γ F attached to some finite Galois extension E of F. Langlands built this action into the dual group as the semidirect product L G E = Ĝ Γ E/F, or more canonically L G = Ĝ Γ F, that is now known as the L-group. If G equals GL(N) for example, Ĝ is just the complex general linear group GL(N, C). In this case, the action of Γ F on Ĝ is trivial, so we can take E = F. For the case that G is orthogonal or symplectic, the families that will be our ultimate interest, we refer the reader to the beginning of 5.
L-functions and Automorphic Representations 13 Automorphic L-functions L(s, π, r) were defined by Langlands for any G. They depend on an automorphic representation π of G and a finite dimensional representation r : L G GL(N, C) (3.2) of L G. It is understood that r is continuous on Γ F and analytic on Ĝ, and in particular, that it factors through a finite quotient Γ E/F of Γ F. Its image is therefore a complex group with finitely many connected components. We will review the definition in the rest of this section. We first recall [F] that any π can be written as a restricted tensor product π = π v, π v Π(G v ), (3.3) v where Π(G v ) is the set of irreducible representations of the locally compact completion G(F v ) of G(F ). The interest is not so much in the individual constituents π v of π, as in the relations they need to satisfy among themselves in order that the product be automorphic. Much of the data that characterize these representations is quite explicit. For example, it is a consequence of what it means for (3.3) to be a restricted direct product that the irreducible representations π v of G(F v ) will be unramified 1 for almost all v. A well known integral transform, introduced into p-adic harmonic analysis by Satake, leads to a canonical mapping 2 π v c(π v ) from the set of unramified representations π v of G(F v ) to the set of semisimple conjugacy classes in L G. The automorphic representation π thus gives rise to a family c S (π) = {c v (π) = c(π v ) : v S} of semisimple classes in L G. If we are given r as well as π, we obtain a family { r ( cv (π) ) : v / S } of semisimple conjugacy classes in GL(N, C). The incomplete automorphic L- function of π and r is then defined in terms of the characteristic polynomials of these classes. It equals the product L S (s, π, r) = v / S L(s, π v, r v ), (3.4) 1 This means that F v is nonarchimedean, that G v = G F F v is quasisplit and split over an unramified extension of F v, and that the restriction of π v to a hyperspecial maximal compact subgroup K v of G(F v) contains the trivial 1-dimensional representation. 2 The mapping becomes a bijection if one takes the restricted form L G E of the L-group, and then takes its range to be the set of semisimple conjugacy classes in L G E whose image in Γ E/F equals the Frobenius class Frob v = Frob E/F,v, if v is unramified in E. This basic condition on the Satake transform was observed by Langlands in [L1].
14 James Arthur where and L(s, π v, r v ) = det ( 1 r ( c v (π) ) qv s ) 1, (3.5) q v = p fv v = o v /p v is the order of the residue class field of F v. The product is easily seen to converge for s in some right half plane, and is clearly a Dirichlet series (1.1) with Euler product (2.1). The definition can be compared with that of the incomplete Artin L-function (2.4) and (2.5), or rather its generalization from Q to F. The analogy is clear, even though the earlier definition was in terms of ideals rather than the formulation here in terms of valuations. Langlands introduced automorphic L-functions in [L1]. He conjectured that they have analytic continuation, with a very precise functional equation L(s, π, r) = ε(s, π, r) L(1 s, π, r ), (3.6) where L(s, π, r) = L S (s, π, r) L S (s, π, r) = v L(s, π v, r v ) (3.7) is a completed L-function obtained by appending a finite product L S (s, π, r) = v S L(s, π v, r v ) of suitable factors at the ramified valuations v S (including the archimedean valuations v S of F ), and ε(s, π, r) = v S ε(s, π v, r v, ψ v ) (3.8) is a finite product of local monomials in qv s. The local ε-factors on the right would depend on the local components ψ v of a fixed, nontrivial additive character ψ on the group F \A, while the global product on the left hand side of (3.8) would be independent of ψ. Langlands did not define the ramified local L- and ε-factors in [L1]. Nevertheless, his introduction of the general automorphic L-function in [L1], with its proposed functional equation (3.6), was an enormous step beyond the abelian automorphic L-functions of Hecke. 3 It depends above all on the L-group L G he introduced at the same time. We have now described two fundamental families of L-functions. They are the arithmetic L-functions of the last section and the automorphic L-functions of this section. Langlands later conjectured that the former family is a subset of the latter. In other words, for any motive M (which for present purposes we can take to be an N-dimensional representation of a finite Galois group Γ E/F ), there should be a pair (π, r) such that the completed L-functions satisfy L(s, M) = L(s, π, r). (3.9) 3 Hecke also introduced some nonabelian L-functions for the group G = GL(2) and the standard two dimensional representation r.
L-functions and Automorphic Representations 15 In particular, the analytic continuation and functional equation for L(s, M) would follow from the same properties for L(s, π, r). This would be a striking and far reaching generalization of the method used by Artin to establish the analytic continuation and functional equation of abelian Artin L-functions. There is actually a theory that applies directly to nonabelian Artin L-functions. Richard Brauer established a general property of the representations of a finite group (the Brauer induction theorem), which he used to express any nonabelian Artin L-function L(s, r) as a quotient of finite products of abelian Artin L-functions L(s, r 1 ) (over finite extensions F 1 of F ). Combined with the results of Artin described in 2, this shows that L(s, r) has analytic continuation, with a functional equation of the desired sort. However, it does not give much control over the analytic behaviour of L(s, r). In particular, it gives no information on a fundamental conjecture of Artin, which asserts that an irreducible, nonabelian Artin L-function is entire. Brauer s theorem has, however, led to important results on local arithmetic L- and ε-factors. These apply more generally to the variant of the Galois group that Weil introduced as a consequence of class field theory. Like the absolute Galois group Γ F, the Weil group W F is defined if F is a local or a global field. It is a locally compact group, equipped with a continuous homomorphism W F Γ F with dense image and connected kernel, whose maximal abelian quotient is given by W ab F = W F /W c F = { F, F \A, if F is local, if F is global. (3.10) If F is the global field we are discussing here, W F comes with a conjugacy class of embeddings W Fv Γ Fv W F for any v, that are compatible with the abelianization (3.10). (See [T3, 1].) These properties imply that the Artin reciprocity law described in 2 extends to a canonical isomorphism from the group of 1-dimensional representations of W F to the group of (1-dimensional) automorphic representations of G = GL(1). Moreover, the Brauer induction theorem extends to N-dimensional representations r of W F. It leads to a global L-function Γ F L(s, r) = v L(s, r v ) = v S L(s, r v ) v / S det ( 1 r(frob v )qv s ) 1 (3.11) that has analytic continuation and functional equation L(s, r) = ε(s, r) L(s, r ), (3.12) for a global ε-factor ε(s, r) = v S ε(s, r v, ψ v ). (3.13)
16 James Arthur As in the special case of Artin L-functions, we obtain little control over the analytic behaviour of the global L-functions L(s, r) in the full complex domain. The global interest in these results is therefore limited. However, Deligne used them to establish important local results [T3, 2]. He showed that the local L- functions L(s, r v ) in (3.11) and ε-factors ε(s, r v, ψ v ) in (3.13) have a canonical local definition. In particular, they can be constructed independently of the global representation r. The global L-functions L(s, r) attached to representations of W F are not all motivic, unlike Artin L-functions. We cannot therefore really regard them as arithmetic. On the other hand, they are not automorphic, since they are not generally defined in terms of automorphic representations. Perhaps they should be regarded as objects that lie between the two classes. In any case, L(s, r) should still be equal to an automorphic L-function. This is again among the original conjectures of Langlands in [L1]. Deligne s constructions then become important for the local classification of representations, and in particular, for comparison with the local L- and ε-factors in (3.7) and (3.8). 4. The principle of functoriality Langlands conjectural functional equation (3.6) for a general automorphic L- function is very deep, and far from known. However, among the cases that are known, there is one that deserves special mention. It is the standard automorphic L-function, in which G equals GL(N), and r equals the standard N-dimensional representation St of L G F = GL(N, C). Abelian Hecke L-functions are given by the further special case that G = GL(1). Hecke established their analytic continuation and functional equation, using the classical language of ideals. Tate later simplified Hecke s proof by introducing the ring of adeles A. In his famous thesis [T1], he established the results through the interplay of multiplicative harmonic analysis on the idele class group F \A = GL(1, F )\GL(1, A) with additive harmonic analysis on the group A. Following Langlands paper [L1], Godement and Jacquet [GJ] extended the method of Tate to GL(N), with the additive group of adelic matrices M N N (A) in place of A. It follows from their results and later refinements [J] that the standard (completed) L-function L(s, π) = L(s, π, St) (4.1) for any automorphic representation π of GL(N) is well defined, and has analytic continuation with functional equation L(s, π) = ε(s, π) L(1 s, π ). This method also gives further information about the analytic behaviour of standard L-functions. For example, if the automorphic representation π is cuspidal,
L-functions and Automorphic Representations 17 L(s, π) is an entire function of s unless N = 1 and π(x) = x λ for some λ C, in which case L(s, π) is analytic apart from a simple pole at s = 1 λ. For any G, we write Π aut (G) for the set of automorphic representations of G (in the broad sense of [L4] we have agreed upon). We then write C aut (G) = {c(π) : π Π aut (G)} (4.2) for the set of families c S (π) of automorphic conjugacy classes, taken up to the equivalence relation defined by c S c S 1 if c v = c 1,v for almost all v. We emphasize that these are concrete objects. They represent the fundamental data encompassed in the seemingly abstract notion of an automorphic representation. As we have noted, the arithmetic significance of these data is not so much in the value of any one class c v (π) as in the relationships among the classes as v varies. In his original paper [L1], Langlands made a profound conjecture that later became known as the principle of functoriality. We shall state it in the restricted form that applies to the concrete families C aut (G). Principle of Functoriality (Langlands). Suppose that G and G are quasisplit 4 groups over the number field F. Suppose also that ρ : L G L G is an L-homomorphism (that is, a continuous, analytic homomorphism that commutes with the two projections onto Γ F ) between their L-groups. Then if c = {c v} lies in C aut (G ), the family c = ρ(c ) = {ρ(c v)} lies in C aut (G). In other words, if c = c(π ) for some π Π aut (G ), then c = c(π) for some π Π aut (G). The principle of functoriality is the central problem in the theory of automorphic forms. It asserts that the internal relations in an automorphic family c = c(π ) for G, whatever they might be, are reflected in the internal relations in some automorphic family c = c(π) for G. The principle of functoriality has been established in a significant number of cases. But as challenging as these have been, they pale in comparison with the cases that have not been established. In the same paper [L1], Langlands pointed out some fundamental applications of functoriality. The first concerns the automorphic L-functions he had just introduced. Suppose that G, G, ρ, c = c(π ) and c = c(π) are as in the assertion of functoriality. If r is an N-dimensional representation of L G, the composition r ρ is an N-dimensional representation of L G. We then obtain an identity L S (s, π, r ρ) = L S (s, π, r) (4.3) of incomplete automorphic L-functions from the definitions, and of course, the principle of functoriality. This relation might seem almost routine at first glance, 4 See the brief description of this property at the beginning of the next section.
18 James Arthur certainly not the sweeping observation it actually is. But consider the special case with G, GL(N), c = c(π) and c N = c(π N ) in place of G, G, c and c respectively. Then ρ can be identified with an N-dimensional representation r of L G, and (4.3) specializes to an identity L S (s, π, r) = L S (s, π N, St) = L S (s, π N ). The general incomplete automorphic L-function on left thus equals a standard incomplete L-function, the function for which we already have analytic continuation and functional equation. If we set 5 and L S (s, π, r) = L S (s, π N ) ε(s, π, r) = ε(s, π N ), for the supplementary terms, the completed L-function satisfies L(s, π, r) = L S (s, π, r)l S (s, π, r) = L S (s, π N )L S (s, π N ) = L(s, π N ), (4.4) and the general functional equation (3.6) then follows from its analogue for standard L-functions. A second immediate application of functoriality pointed out by Langlands in [L1] is to nonabelian class field theory. It concerns the seemingly trivial case of functoriality with G = {1}. Despite its apparent simplicity, however, this case comes with answers to the two general questions (I) and (II) from the preface. If G equals {1}, the dual group Ĝ also equals {1}, but the L-group L G is still the absolute Galois group Γ F. We again take the second group G to be GL(N). An L-homomorphism from L G to L G will be continuous (by definition) on its totally disconnected domain Γ F. It can therefore be identified with an N-dimensional representation r : Γ E/F GL(N, C) (4.5) of the Galois group of some finite Galois extension E of F. The only automorphic representation of G is the trivial representation 1. However, the associated automorphic family c(1) is still interesting. It is represented by the set c S (1) = {c v (1) = Frob v : v / S} of Frobenius conjugacy classes in Γ E/F of primes v of F that are unramified in E, according to footnote 2 from the last section. The principle of functoriality in this 5 In order that the two left hand sides here depend only on π, we assume implicitly that π N is isobaric, in the sense of [L5, 2]. It is then the unique automorphic representation of GL(N) with the given eigenfamily c(π N ) = c(π), according to [JS]. Notice that we do not obtain a local construction for the factors in these supplementary terms, unlike in their analogues (3.11), (3.13) for representations of Weil groups. This requires a stronger (and more complex) assertion of functoriality as in [L1], and is predicated on a local classification of representations, such as that obtained for special orthogonal and symplectic groups in [A4].
L-functions and Automorphic Representations 19 case asserts that its r-image r ( c(1) ) is automorphic for GL(N). In other words, there is an automorphic representation π of GL(N) such that c v (π) = r ( c v (1) ) = r(frob v ), v / S. This can be regarded as a general answer to the question (I). Since it includes an analytic characterization of the set Spl(E/F ) of primes of F that split completely in E, it also amounts to a classification theory for general Galois extensions of F, the long standing dream of earlier number theorists. The arithmetic data {Frob v } that characterize finite Galois extensions E of F ([T2, p. 165]), and that are conveniently packaged by the characters of continuous, finite dimensional representations of Γ F, can be represented by the analytic data {c v (π)} of automorphic representations of general linear groups. Langlands formulation of nonabelian class field theory has implications for L-functions. It follows from the definitions (2.4), (2.5), (3.5) and (3.4) that for G = {1}, the automorphic L-function L(s, 1, r) on the left hand side of (4.4) is equal to the completed Artin L-function L(s, r). It therefore equals an automorphic L-function L(s, π N ) = L(s, π) for GL(N). We should note here that the general principle of functoriality implicitly includes some common spectral properties of the two automorphic representations π and π. In particular, if the representation r in (4.4) is irreducible, the automorphic representation π N of GL(N) should be cuspidal. If N 2, this implies that the automorphic L-function L(s, π N ) is entire, as we noted at the beginning of the section. On the other hand, the Artin conjecture mentioned near the end of 3 asserts that the irreducible L- function L(s, r) is entire. The principle of functoriality, in the case G = {1} and G = GL(N), therefore implies this well known conjecture of almost one hundred years. By relating L(s, r) to a standard automorphic L-function L(s, π N ), we would obtain what could be considered an answer to the question (II). For since we now have an understanding of the poles of L(s, π N ), and can perhaps hope someday to have also an understanding of its zeros, we would have the means to estimate the distribution of the classes {Frob v }. Notice that this is a beautiful generalization of the indirect method in Section 2 used by Artin to study abelian L-functions. In both cases, an analytic problem for arithmetic L-functions is solved by showing that these functions are also automorphic L-functions, for which the analytic behaviour is better understood. In addition to the two striking consequences of functoriality in [L1] we have just described, Langlands proposed two further applications. One is to the generalized Ramanujan conjecture, the other to a generalization of the conjecture of Sato-Tate. Both have implications for L-functions. For the sake of completeness, we shall say a word on each. The generalized Ramanujan conjecture can be formulated for any G. It asserts that the local constituents π v of certain natural automorphic representations π in the discrete spectrum of G are (locally) tempered, in the extension to the local groups G(F v ) by Harish-Chandra of the definition from classical Fourier analysis. If G = GL(N), for example, it is the unitary cuspidal automorphic representations to which the conjecture applies. The connection with L-functions, suggested
20 James Arthur by Langlands and reinforced by later local harmonic analysis, is that the local components π v of π will be tempered if and only if the L-functions L(s, π, r) are analytic in the right half plane Re(s) > 1. Given this property, Langlands deduced the generalized Ramanujan conjecture from the principle of functoriality, and the fact that a representation π in the discrete spectrum is automatically unitary [L1, p. 43 49]. For more recent progress for the group G = GL(2), we refer the reader to [KiS] and [Ki]. A generalized Sato-Tate conjecture is only hinted at in [L1], and then just in the last paragraph on p. 49. It would apply to any automorphic representation π of G that satisfies the generalized Ramanujan conjecture. We assume the principle of functoriality. The Ramanujan condition is then valid. It implies that for any unramified place v / S, the conjugacy class c v (π) in L G intersects a fixed maximal compact subgroup L K = K Γ F in L G = Ĝ Γ F, and can therefore be identified with a conjugacy class in L K. The problem is to determine the distribution of these classes as v varies. If the family c(π) is a proper functorial image of a family c(π ) for some group G, one could determine the distribution of c(π) from that of c(π ). One can therefore assume that π is primitive, in the sense that it is not a proper functorial image from some G. We would then expect 6 that ord s=1 ( L(s, π, r) ) = [r : 1] for any finite dimensional representation ρ of L G. That is, the order of the pole of L(s, π, r) at s = 1 equals the multiplicity of the trivial representation of L G in r. It would then follow from the Wiener-Ikehara Tauberian theorem that the distribution of the classes {c v (π)} in L K is given by the Haar measure on L K. In concrete terms, one would be able to express the distribution of classes {c v (π) L T : v / S} in a maximal torus L T = T Γ F in L K in terms of the explicit density function on L T obtained from the Haar measure that occurs in the Weyl integration formula. (See [Se, 2, Appendix].) I have attempted to introduce the subject through Langlands original paper [L1] without discussing subsequent developments. The most famous of these is undoubtedly Wiles work on the Shimura-Taniyama-Weil conjecture [Wi], which he used to prove Fermat s Last Theorem. An important foundation for Wiles work was the Langlands-Tunnell theorem that two-dimensional representations r of solvable Galois groups Γ E/F satisfy Artin s conjecture. This followed from base change for GL(2) [L6], established by Langlands as an early application of the trace formula. (See [A1, 26], for example.) I mention also that R. Taylor has established the classical Sato-Tate conjecture, which applies to the group G = GL(2), by using base change for GL(N) [AC] and other means to extract what is needed from the unproven principle of functoriality. (See [Tay] and the references there.) 6 This is actually a little stronger than the principle of functoriality, of which it represents a converse of sorts. However, Langlands recent ideas [L7] for attacking the principle of functoriality, speculative as they may be, would treat this question as well.
L-functions and Automorphic Representations 21 5. Orthogonal and symplectic groups The monograph [A4] contains a classification of automorphic representations of quasisplit orthogonal and symplectic groups over the number field F. The groups of interest are attached to the four infinite families of complex simple Lie algebras. These in turn are represented by the following four infinite families of Coxeter- Dynkin diagrams, for which I am indebted to W. Casselman. Type A n Type B n Type C n Type D n For corresponding complex groups, we could take the special linear groups SL(n + 1, C), the odd orthogonal groups SO(2n + 1, C), the symplectic groups Sp(2n, C) and the even orthogonal groups SO(2n, C). The family A n is the starting point for the classification. Since the representation theory is simplest for general linear groups [JS], [MW1], we take the reductive groups GL(N, C), N = n+1, as the complex representatives for this family. We want to take these groups over the number field F, which is not algebraically closed. But according to a fundamental theorem of Chevalley, any one of these complex groups corresponds naturally to a canonical group over F. It is the split group attached to the given diagram (and centre). Our interest is actually in quasisplit groups. These are obtained by twisting the Galois action on any given split group by a supplementary Galois action on the diagram. The symmetry group of a diagram is the group of bijections of the set of vertices that preserve all edges and directions. It equals Z/2Z in type A n, is trivial in types B n and C n, and equals 7 Z/2Z in type D n. A quasisplit group is determined by a homomorphism from the Galois group Γ F to the symmetry group of the diagram. Following [A5], we will not treat nonsplit, quasisplit groups of type A n. These are unitary groups, for which we refer the reader to [Mok]. Since a quasisplit group of type B n or C n is split, we have only then to consider type D n. In this case, a quasisplit group is determined by a quotient of Γ F of order 1 or 2, or in other words, a Galois extension E/F of degree 1 or 2. 7 If n = 4, this group is actually isomorphic to S 3, but we agree to consider only the standard symmetries that interchange the two right hand vertices in the diagram.